Consider an asymmetric tent design of length l that is

Question

Sample midterm 1 questions-

1. By using appropriate series tests, determine whether

_n=0Σ^∞(1/4ⁿ + (1/2)ⁿ,          _n=1Σ^∞sin nπ/4,    _n=1Σ^∞(1/n log n)

converge or diverge.

2. (a) Calculate the exact interval of convergence of

_n=0Σ^∞ yⁿ/3ⁿ.

(b) Sketch the function f(x) = 3 + (x² - 1)(x² - 4) over the range -3 < x < 3.

(c) Determine the exact ranges of convergence of

_n=0Σ^∞ f(x)ⁿ/3ⁿ.

3. Consider a right-angled triangle with vertices at (0, 0), (0, 1), and (1, 0), with non-uniform density ρ(x, y) = y. Find its center of mass.

4. Find the minimum and maximum values of the function f(x, y) = (x - 2y)² - x in the square |x| ≤ 1, |y| ≤ 1.

5. Consider an asymmetric tent design of length l that is comprised of a vertical section of height h, connected to a diagonal section, as shown in the diagram below.

The volume of the tent is hl²/2. For a fixed area of tent material A, find the values of h and l that maximise the tent's volume.